Yes, it'll have the same meaning, and it's a perfectly natural way to express that you are neither a doctor nor a lawyer. I'm neither a doctor nor a lawyer. I'm not a doctor or a lawyer. For the example where 'neither' is used first, it's more correct to use 'nor' in place of 'or.' If you use 'not' first, then 'or' is correct. Generally, in the US, it's fairly common to hear people not using 'nor' at all in more casual speech.